Three of the following four number-pairs are alike in a certain way and thus form a group. Which number-pair does NOT belong to that group? (NOTE: The relation should be found without breaking down the numbers into its constituent digits)
- (a)31, 41
- (b)73, 79
- (c)83, 97
- (d)61,71
Answer
Why
Correct — B. All eight numbers are prime, so primality by itself cannot separate them. The separating property is what sits between the two.
Rule: the second number is the second prime after the first — one prime is skipped.
31 → 37 → 41
83 → 89 → 97
61 → 67 → 71
73 and 79 are consecutive primes: 74 to 78 contain none. Option (b) skips nothing, so it is the pair that leaves the group.
Why the others are wrong
- (a)31, 41 — 31, 41 skips 37, exactly what the group does. The gap of 10 is what makes it look like a candidate, but that gap is shared with 61, 71.
- (c)83, 97 — 83, 97 draws the eye because its gap is 14, the widest here. It still skips one prime, 89, so it belongs with the group.
- (d)61,71 — 61, 71 skips 67 and fits the rule. Pairing it with 31, 41 makes a difference of 10 look like the rule, and 83, 97 then breaks it.
Concept
Number-pair odd-one-out items carry a rider — do not break the numbers into digits — precisely because the intended relation is arithmetic on the whole number.
When every number is odd and none is a familiar square or cube, test primality first. If all of them are prime, the rule has to be about the gap between them, measured in primes rather than in units.
Counting in primes is the step most candidates skip: 31 to 41 passes one prime, and so does 61 to 71 and 83 to 97.
The difference between the numbers looks like a rule at first, since 31, 41 and 61, 71 are both 10 apart. It fails on 83, 97, which is 14 apart, and a rule that gathers only two of the four cannot be the intended one.
Key facts
- The primes in play are 31, 37, 41, 61, 67, 71, 73, 79, 83, 89 and 97.
- 73 and 79 are consecutive primes: 74, 75, 76, 77 (7 × 11) and 78 are all composite.
- The rider printed with this question forbids splitting a number into its digits, which rules out digit sums and reversals.
Study next
Common traps
- Fixing on the difference of 10 without checking that it gathers three of the four pairs.
- Choosing 83, 97 because its gap is widest.
- Splitting the numbers into digits despite the printed rider.
Whole-number relations recur across the section: Reasoning Q.2 asks for a triad matching 16-40-100 and 8-20-50, Reasoning Q.10 relates 16 to 30 and 81 to 110, and Reasoning Q.24 hangs on a constant difference of 22.
Related PYQs
No directly related past PYQ was found.